Mathieu Groups

The first Simple Sporadic Groups discovered. $M_{11}$, $M_{12}$, $M_{22}$, $M_{23}$, $M_{24}$ were discovered in 1861 and 1873 by Mathieu. Frobenius showed that all the Mathieu groups are Subgroups of $M_{24}$.

The Mathieu groups are most simply defined as Automorphism groups of Steiner Systems. $M_{11}$ corresponds to $S(4,5,11)$ and $M_{23}$ corresponds to $S(4,7,23)$. $M_{11}$ and $M_{23}$ are Transitive Permutation Groups of 11 and 23 elements.

The Orders of the Mathieu groups are

$$\begin{eqnarray*} \vert M_{11}\vert&=&2^4\cdot 3^2\cdot 5\cdot 11\\ \vert M_{... ...ert M_{24}\vert&=&2^{10}\cdot 3^3\cdot 5\cdot 7\cdot 11\cdot 23. \end{eqnarray*}$$

See also Sporadic Group

References

Conway, J. H. and Sloane, N. J. A. ``The Golay Codes and the Mathieu Groups.'' Ch. 11 in Sphere Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 299-330, 1993.

Rotman, J. J. Ch. 9 in An Introduction to the Theory of Groups, 4th ed. New York: Springer-Verlag, 1995.

Wilson, R. A. ``ATLAS of Finite Group Representation.'' http://for.mat.bham.ac.uk/atlas/html/contents.html#spo.